Estimating R0 from coalescent growth rates

The basic reproduction number of the swine influenza epidemic, \( R_{0} \) can be estimated from its initial rate of spread. If we assume roughly exponential growth then the basic reproductive number is related to the growth rate by the so-called Lotka-Euler estimating equation:

\[ R_{0}=\int_{0}^{\infty} \frac{1}{w(\tau)e^{-rt}} d\tau \]

where \( r \) is the rate of exponential growth and \( w(\tau) \) the generation time distribution [1]. The generation time distribution can be thought of as the probability density function describing the distribution of times between successive infections in a chain of transmission.

The estimate of the basic reproductive number is therefore dependent not just on an estimate of \( r \) , but also a good estimate of the generation time distribution [2]. In the case of swine influenza the generation time distribution is unclear, but data appear consistent with seasonal influenza that has a mean generation time of approximately 3 days.

Analytical solutions for \( R_{0} \) can be derived for different assumed generation time distributions using the Lotka-Euler estimating equation (which is essentially a moment generating function). If we assume a generation time distribution that follows the gamma distribution, then

\[ R_{0}=\left(1+\frac{r}{b}\right)^{a} \]

where a and b are the parameters of the gamma distribution (\( a = m^{2}/s^{2} \) and \( b = m/s^{2} \) where \( m \) and \( s \) are the mean and standard deviation of the distribution respectively).

Estimates of \( R_{0} \) based on the estimates of \( r \) reported by Andrew Rambaut are given in Table 1. Obviously \( r \) can also be estimated from epidemiological case data and this may give different results.

\( r \) (per day with 95% HPD)

generation time distribution (and parameters in days)

\( R_{0} \)

0.053 (0.0014, 0.12)

gamma (\( m=3 \), \( s=2 \))

1.17 (1.00 - 1.40)

0.053 (0.0014, 0.12)

exponential (\( m=3 \)) (i.e. SIR model)

1.16 (1.00 - 1.36)

Table 1 | Estimates of \( R_{0} \) from the coalescent growth rate, \( r \) for the early period of Pandemic H1N1.